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Magnetic systems, in which due to competition between the direct Heisenberg exchange and the Dzyaloshinskii –Moriya interaction, magnetic vortex structures — skyrmions appear, were studied using the Metropolis algorithm.

The conditions for the nucleation and stable existence of magnetic skyrmions in two-dimensional magnetic films in the frame of the classical Heisenberg model were considered in the article. A thermal stability of skyrmions in a magnetic film was studied. The processes of the formation of various states in the system at different values of external magnetic fields were considered, various phases into which the Heisenberg spin system passes were recognized. The authors identified seven phases: paramagnetic, spiral, labyrinth, spiralskyrmion, skyrmion, skyrmion-ferromagnetic and ferromagnetic phases, a detailed analysis of the configurations is given in the article.

Two phase diagrams were plotted: the first diagram shows the behavior of the system at a constant $D$ depending on the values of the external magnetic field and temperature $(T, B)$, the second one shows the change of the system configurations at a constant temperature $T$ depending on the magnitude of the Dzyaloshinskii – Moriya interaction and external magnetic field: $(D, B)$.

The data from these numerical experiments will be used in further studies to determine the model parameters of the system for the formation of a stable skyrmion state and to develop methods for controlling skyrmions in a magnetic film.

This article solves the problem of developing a technology for collecting initial data for building models for assessing the functional state of a person. This condition is assessed by the pupil response of a person to a change in illumination based on the pupillometry method. This method involves the collection and analysis of initial data (pupillograms), presented in the form of time series characterizing the dynamics of changes in the human pupils to a light impulse effect. The drawbacks of the traditional approach to the collection of initial data using the methods of computer vision and smoothing of time series are analyzed. Attention is focused on the importance of the quality of the initial data for the construction of adequate mathematical models. The need for manual marking of the iris and pupil circles is updated to improve the accuracy and quality of the initial data. The stages of the proposed technology for collecting initial data are described. An example of the obtained pupillogram is given, which has a smooth shape and does not contain outliers, noise, anomalies and missing values. Based on the presented technology, a software and hardware complex has been developed, which is a collection of special software with two main modules, and hardware implemented on the basis of a Raspberry Pi 4 Model B microcomputer, with peripheral equipment that implements the specified functionality. To evaluate the effectiveness of the developed technology, models of a single-layer perspetron and a collective of neural networks are used, for the construction of which the initial data on the functional state of intoxication of a person were used. The studies have shown that the use of manual marking of the initial data (in comparison with automatic methods of computer vision) leads to a decrease in the number of errors of the 1st and 2nd years of the kind and, accordingly, to an increase in the accuracy of assessing the functional state of a person. Thus, the presented technology for collecting initial data can be effectively used to build adequate models for assessing the functional state of a person by pupillary response to changes in illumination. The use of such models is relevant in solving individual problems of ensuring transport security, in particular, monitoring the functional state of drivers.

In this paper, the system of equations of magnetic hydrodynamics (MHD) is considered. The exact solutions found describe fluid flows in a porous medium and are related to the development of a core simulator and are aimed at creating a domestic technology «digital deposit» and the tasks of controlling the parameters of incompressible fluid. The central problem associated with the use of computer technology is large-dimensional grid approximations and high-performance supercomputers with a large number of parallel microprocessors. Kinetic methods for solving differential equations and methods for «gluing» exact solutions on coarse grids are being developed as possible alternatives to large-dimensional grid approximations. A comparative analysis of the efficiency of computing systems allows us to conclude that it is necessary to develop the organization of calculations based on integer arithmetic in combination with universal approximate methods. A class of exact solutions of the Navier – Stokes system is proposed, describing three-dimensional flows for an incompressible fluid, as well as exact solutions of nonstationary three-dimensional magnetic hydrodynamics. These solutions are important for practical problems of controlled dynamics of mineralized fluids, as well as for creating test libraries for verification of approximate methods. A number of phenomena associated with the formation of macroscopic structures due to the high intensity of interaction of elements of spatially homogeneous systems, as well as their occurrence due to linear spatial transfer in spatially inhomogeneous systems, are highlighted. It is fundamental that the emergence of structures is a consequence of the discontinuity of operators in the norms of conservation laws. The most developed and universal is the theory of computational methods for linear problems. Therefore, from this point of view, the procedures of «immersion» of nonlinear problems into general linear classes by changing the initial dimension of the description and expanding the functional spaces are important. Identification of functional solutions with functions makes it possible to calculate integral averages of an unknown, but at the same time its nonlinear superpositions, generally speaking, are not weak limits of nonlinear superpositions of approximations of the method, i.e. there are functional solutions that are not generalized in the sense of S. L. Sobolev.

When modeling turbulent flows in practical applications, it is often necessary to carry out a series of calculations of bodies of similar topology. For example, bodies that differ in the shape of the fairing. The use of convolutional neural networks allows to reduce the number of calculations in a series, restoring some of them based on calculations already performed. The paper proposes a method that allows to apply a convolutional neural network regardless of the method of constructing a computational mesh. To do this, the flow field is reinterpolated to a uniform mesh along with the body itself. The geometry of the body is set using the signed distance function and masking. The restoration of the flow field based on part of the calculations for similar geometries is carried out using a neural network of the UNet type with a spatial attention mechanism. The resolution of the nearwall region, which is a critical condition for turbulent modeling, is based on the equations obtained in the nearwall domain decomposition method.

A demonstration of the method is given for the case of a flow around a rounded plate by a turbulent air flow with different rounding at fixed parameters of the incoming flow with the Reynolds number $Re = 10^5$ and the Mach number $M = 0.15$. Since flows with such parameters of the incoming flow can be considered incompressible, only the velocity components are studied directly. The flow fields, velocity and friction profiles obtained by the surrogate model and numerically are compared. The analysis is carried out both on the plate and on the rounding. The simulation results confirm the prospects of the proposed approach. In particular, it was shown that even if the model is used at the maximum permissible limits of its applicability, friction can be obtained with an accuracy of up to 90%. The work also analyzes the constructed architecture of the neural network. The obtained surrogate model is compared with alternative models based on a variational autoencoder or the principal component analysis using radial basis functions. Based on this comparison, the advantages of the proposed method are demonstrated.

X-ray diffraction experiment allows determining of magnitudes of complex coefficients in the decomposition of the studied electron density distribution into Fourier series. The determination of the lost in the experiment phase values poses the central problem of the method, namely the phase problem. Some methods for solving of the phase problem result in a set of feasible solutions. Cluster analysis method may be used to investigate the composition of this set and to extract one or several typical solutions. An essential feature of the approach is the estimation of the closeness of two solutions by the map correlation between two aligned Fourier syntheses calculated with the use of phase sets under comparison. An interactive computer program ClanGR was designed to perform this analysis.

Technical diagnosis’ current trends are connected to application of FEM computer simulation, which allows, to some extent, replace real experiments, reduce costs for investigation and minimize risks. Computer simulation, just at the stage of research and development, allows carrying out of diagnostics of equipment to detect permissible fluctuations of parameters of equipment’s work. Peculiarity of diagnosis of rolling equipment is that functioning of rolling equipment is directly tied with manufacturing of product with required quality, including accuracy. At that design of techniques of technical diagnosis and diagnostical modelling is very important. Computer simulation of cold rolling of strip was carried out. At that upper working roll was doing vibrations in horizontal direction according with published data of experiments on continuous 1700 rolling mill. Vibration of working roll in a stand appeared due to gap between roll’s craft and guide in a stand and led to periodical fluctuations of strip’s thickness. After computer simulation with the help of DEFORM software strip with longitudinal and transversal thickness variation was gotten. Visualization of strip’s geometrical parameters, according with simulation data, corresponded to type of inhomogeneity of surface of strip rolled in real. Further analysis of thickness variation was done in order to identify, on the basis of simulation, sources of periodical components of strip’s thickness, whose reasons are malfunctions of equipment. Advantage of computer simulation while searching the sources of forming of thickness variation is that different hypothesis concerning thickness formations may be tested without conducting real experiments and costs of different types may be reduced. Moreover, while simulation, initial strip’s thickness will not have fluctuations as opposed to industrial or laboratorial experiments. On the basis of spectral analysis of random process, it was established that frequency of changing of strip’s thickness after rolling in one stand coincides with frequency of working roll’s vibration. Results of computer simulation correlate with results of the researches for 1700 mill. Therefore, opportunity to apply computer simulation to find reasons of formation of thickness variation of strip on the industrial rolling mill is shown.

The article deals with the process of creating an effective algorithm for determining the amount of emitted quanta from an X-ray tube in computer tomography (CT) studies. An analysis of domestic and foreign literature showed that most of the work in the field of radiometry and radiography takes the tabulated values of X-ray absorption coefficients into account, while individual dose factors are not taken into account at all since many studies are lacking the Dose Report. Instead, an average value is used to simplify the calculation of statistics. In this regard, it was decided to develop a method to detect the amount of ionizing quanta by analyzing the noise of CT data. As the basis of the algorithm, we used Poisson and Gauss distribution mathematical model of owns’ design of logarithmic value. The resulting mathematical model was tested on the CT data of a calibration phantom consisting of three plastic cylinders filled with water, the X-ray absorption coefficient of which is known from the table values. The data were obtained from several CT devices from different manufacturers (Siemens, Toshiba, GE, Phillips). The developed algorithm made it possible to calculate the number of emitted X-ray quanta per unit time. These data, taking into account the noise level and the radiuses of the cylinders, were converted to X-ray absorption values, after which a comparison was made with tabulated values. As a result of this operation, the algorithm used with CT data of various configurations, experimental data were obtained, consistent with the theoretical part and the mathematical model. The results showed good accuracy of the algorithm and mathematical apparatus, which shows reliability of the obtained data. This mathematical model is already used in the noise reduction program of the CT of own design, where it participates as a method of creating a dynamic threshold of noise reduction. At the moment, the algorithm is being processed to work with real data from computer tomography of patients.

The traditional approach to computing optimal game strategies of firms on oligopolistic markets and of indicators of such markets consists in studying linear dynamical games with quadratic criteria and solving generalized matrix Riccati equations.

The other approach proposed by the author is based on methods of operational calculus (in particular, Z-transform). This approach makes it possible to achieve economic meaningful decisions under wider field of parameter values. It characterizes by simplicity of computations and by necessary for economic analysis visibility. One of its advantages is that in many cases important for economic practice, it, in contrast to the traditional approach, provides the ability to make calculations using widespread spreadsheets, which allows to study the prospects for the development of oligopolistic markets to a wide range of professionals and consumers.

The article deals with the practical aspects of determining the optimal Nash–Cournot strategies of participants in oligopolistic markets on the basis of operational calculus, in particular the technique of computing the optimal Nash–Cournot strategies in Excel. As an illustration of the opportinities of the proposed methods of calculation, examples close to the practical problems of forecasting indicators of the markets of high-tech products are studied.

The results of calculations obtained by the author for numerous examples and real economic systems, both using the obtained relations on the basis of spreadsheets and using extended Riccati equations, are very close. In most of the considered practical problems, the deviation of the indicators calculated in accordance with the two approaches, as a rule, does not exceed 1.5–2%. The highest value of relative deviations (up to 3–5%) is observed at the beginning of the forecasting period. In typical cases, the period of relatively noticeable deviations is 3–5 moments of time. After the transition period, there is almost complete agreement of the values of the required indicators using both approaches.

The article is devoted to the construction and investigation of an averaged mathematical model of an aluminum-cobalt-molybdenum hydrocracking catalyst oxidative regeneration. The oxidative regeneration is an effective means of restoring the activity of the catalyst when its granules are coating with coke scurf.

The mathematical model of this process is a nonlinear system of ordinary differential equations, which includes kinetic equations for reagents’ concentrations and equations for changes in the temperature of the catalyst granule and the reaction mixture as a result of isothermal reactions and heat transfer between the gas and the catalyst layer. Due to the heterogeneity of the oxidative regeneration process, some of the equations differ from the standard kinetic ones and are based on empirical data. The article discusses the scheme of chemical interaction in the regeneration process, which the material balance equations are compiled on the basis of. It reflects the direct interaction of coke and oxygen, taking into account the degree of coverage of the coke granule with carbon-hydrogen and carbon-oxygen complexes, the release of carbon monoxide and carbon dioxide during combustion, as well as the release of oxygen and hydrogen inside the catalyst granule. The change of the radius and, consequently, the surface area of coke pellets is taken into account. The adequacy of the developed averaged model is confirmed by an analysis of the dynamics of the concentrations of substances and temperature.

The article presents a numerical experiment for a mathematical model of oxidative regeneration of an aluminum-cobalt-molybdenum hydrocracking catalyst. The experiment was carried out using the Kutta–Merson method. This method belongs to the methods of the Runge–Kutta family, but is designed to solve stiff systems of ordinary differential equations. The results of a computational experiment are visualized.

The paper presents the dynamics of the concentrations of substances involved in the oxidative regeneration process. A conclusion on the adequacy of the constructed mathematical model is drawn on the basis of the correspondence of the obtained results to physicochemical laws. The heating of the catalyst granule and the release of carbon monoxide with a change in the radius of the granule for various degrees of initial coking are analyzed. There are a description of the results.

In conclusion, the main results and examples of problems which can be solved using the developed mathematical model are noted.

This work analyzes the problem of traffic signs recognition in intelligent transport systems. The basic concepts of computer vision and image recognition tasks are considered. The most effective approach for solving the problem of analyzing and recognizing images now is the neural network method. Among all kinds of neural networks, the convolutional neural network has proven itself best. Activation functions such as Relu and SoftMax are used to solve the classification problem when recognizing traffic signs. This article proposes a technology for recognizing traffic signs. The choice of an approach for solving the problem based on a convolutional neural network due to the ability to effectively solve the problem of identifying essential features and classification. The initial data for the neural network model were prepared and a training sample was formed. The Google Colaboratory cloud service with the external libraries for deep learning TensorFlow and Keras was used as a platform for the intelligent system development. The convolutional part of the network is designed to highlight characteristic features in the image. The first layer includes 512 neurons with the Relu activation function. Then there is the Dropout layer, which is used to reduce the effect of overfitting the network. The output fully connected layer includes four neurons, which corresponds to the problem of recognizing four types of traffic signs. An intelligent traffic sign recognition system has been developed and tested. The used convolutional neural network included four stages of convolution and subsampling. Evaluation of the efficiency of the traffic sign recognition system using the three-block cross-validation method showed that the error of the neural network model is minimal, therefore, in most cases, new images will be recognized correctly. In addition, the model has no errors of the first kind, and the error of the second kind has a low value and only when the input image is very noisy.